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The range of a cannon is given by the formula $$ \Delta x = \frac{v_i^2}{g} \sin \left( 2 \theta \right) $$ in which \( v_i \) is the initial speed of the cannon, \( g \) is free-fall acceleration, and \( \theta \) is the launch angle of the cannon. The goal of this lab is to:

- Prove that \( \Delta x \) is directly proportional to \( \sin \left( 2 \theta \right) \).
- Use this relationship to determine the initial velocity of a spring-loaded cannon.

However...here is the challenge. This relationship applies only when the cannonball lands on exactly the same vertical level it was launched! Otherwise, there is a much more challenging solution involving the quadratic equation, which we will consider during the lab called "aim a cannon at any point in 3-D space." For the purposes of this lab, we need the cannonball (a marble) to land on the same vertical level it was launched at. To this end, at approximately the point it will land, place 6 physics textbooks in a stack.

**
Important Rule:
**
Whenever firing the cannon,
somebody **must**
be standing outside the range of the cannon, like a baseball catcher,
in a position to catch the cannonball (which is a marble).
It don't want any of mine rolling around the place!

We will not consider air resistance in our calculations.

Compile at least 30 measurements that relate \( \theta \) to \( \Delta x \).
Complete ten measurements on strength 1, ten measurements on strength level 2,
and ten measurements on strength level 3.
Use your data to create the following table (which should be located
in your

Please enter your data into a `.csv`

file
and give it to Mr. Kuncik.
He will produce a scatter plot and several statistical results for this lab.
This will lead to a much larger set of information to draw from.

For example, if the first three measurements were:

Please format your `.csv`

file as follows:

```
strength, theta, range
```

1, 30, 0.85

1, 45, 1.2

2, 35, 1.5